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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
19 minutes ago
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
19 minutes ago
0 replies
Bijective quartic modulo p
DottedCaculator   12
N 33 minutes ago by MathLuis
Source: ELMO 2024/6
For a prime $p$, let $\mathbb{F}_p$ denote the integers modulo $p$, and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$. Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$, there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$. (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.)

Aprameya Tripathy
12 replies
2 viewing
DottedCaculator
Jun 21, 2024
MathLuis
33 minutes ago
No More than √㏑x㏑㏑x Digits
EthanWYX2009   3
N an hour ago by MathisWow
Source: 2024 April 谜之竞赛-3
Let $f(x)\in\mathbb Z[x]$ have positive integer leading coefficient. Show that there exists infinte positive integer $x,$ such that the number of digit that doesn'r equal to $9$ is no more than $\mathcal O(\sqrt{\ln x\ln\ln x}).$

Created by Chunji Wang, Zhenyu Dong
3 replies
EthanWYX2009
Mar 24, 2025
MathisWow
an hour ago
Sum of points' powers
Suntafayato   2
N an hour ago by fadhool
Given 2 circles $\omega_1, \omega_2$, find the locus of all points $P$ such that $\mathcal{P}ow(P, \omega_1) + \mathcal{P}ow(P, \omega_2) = 0$ (i.e: sum of powers of point $P$ with respect to the two circles $\omega_1, \omega_2$ is zero).
2 replies
Suntafayato
Mar 24, 2020
fadhool
an hour ago
IMO ShortList 2001, number theory problem 3
orl   10
N 2 hours ago by OronSH
Source: IMO ShortList 2001, number theory problem 3
Let $ a_1 = 11^{11}, \, a_2 = 12^{12}, \, a_3 = 13^{13}$, and $ a_n = |a_{n - 1} - a_{n - 2}| + |a_{n - 2} - a_{n - 3}|, n \geq 4.$ Determine $ a_{14^{14}}$.
10 replies
1 viewing
orl
Sep 30, 2004
OronSH
2 hours ago
If it is an integer then perfect square
Ecrin_eren   0
3 hours ago


"Let a, b, c, d be non-zero digits, and let abcd and dcba represent four-digit numbers.

Show that if the number abcd / dcba is an integer, then that integer is a perfect square."



0 replies
Ecrin_eren
3 hours ago
0 replies
Sum of arctan
Ecrin_eren   1
N 3 hours ago by Shan3t


Find the value of the sum:
sum from n = 0 to infinity of arctan(k / (n² + kn + 1))


1 reply
Ecrin_eren
3 hours ago
Shan3t
3 hours ago
Inequality
Ecrin_eren   0
3 hours ago


Let a, b, c be positive real numbers. Prove the inequality:

sqrt(a² - ab + b²) + sqrt(b² - bc + c²) ≥ sqrt(a² + ac + c²)



0 replies
Ecrin_eren
3 hours ago
0 replies
Cool vieta sum
Kempu33334   6
N 5 hours ago by Lankou
Let the roots of \[\mathcal{P}(x) = x^{108}+x^{102}+x^{96}+2x^{54}+3x^{36}+4x^{24}+5x^{18}+6\]be $r_1, r_2, \dots, r_{108}$. Find \[\dfrac{r_1^6+r_2^6+\dots+r_{108}^6}{r_1^6r_2^6+r_1^6r_3^6+\dots+r_{107}^6r_{108}^6}\]without Newton Sums.
6 replies
Kempu33334
Yesterday at 11:44 PM
Lankou
5 hours ago
đề hsg toán
akquysimpgenyabikho   3
N 6 hours ago by Lankou
làm ơn giúp tôi giải đề hsg

3 replies
akquysimpgenyabikho
Apr 27, 2025
Lankou
6 hours ago
A problem with a rectangle
Raul_S_Baz   13
N Today at 4:38 PM by undefined-NaN
On the sides AB and AD of the rectangle ABCD, points M and N are taken such that MB = ND. Let P be the intersection of BN and CD, and Q be the intersection of DM and CB. How can we prove that PQ || MN?
IMAGE
13 replies
Raul_S_Baz
Apr 26, 2025
undefined-NaN
Today at 4:38 PM
Find the domain and range of $f(x)=2-|x-5|.$
Vulch   1
N Today at 12:13 PM by Mathzeus1024
Find the domain and range of $f(x)=2-|x-5|.$
1 reply
Vulch
Today at 2:07 AM
Mathzeus1024
Today at 12:13 PM
nice problem
teomihai   1
N Today at 11:58 AM by Royal_mhyasd
Let set $A =\{0,1,2,3,...,n\}$ , where $n$ it is positiv ,integer number.
How many subsets of A contain at least one odd number?
1 reply
teomihai
Today at 11:46 AM
Royal_mhyasd
Today at 11:58 AM
(14n+25)/(2n+1) 'is a perfect square - Portugal OPM 2017 p1
parmenides51   4
N Today at 10:03 AM by Namisgood
Determine all integer values of n for which the number $\frac{14n+25}{2n+1}$ 'is a perfect square.
4 replies
parmenides51
May 15, 2024
Namisgood
Today at 10:03 AM
Inequalities
sqing   4
N Today at 9:46 AM by sqing
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq 4\left(\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{32}{9}\left(\frac{a+b}{b+c}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq  \frac{8}{3}\left(  \frac{a+b}{b+c}+ \frac{b+c}{c+a}+ \frac{c+a}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left(  \frac{a^2+bc}{b^2+ca}+\frac{b^2+ca}{c^2+ab}+\frac{c^2+ab}{a^2+bc}\right)$$
4 replies
sqing
Today at 12:20 AM
sqing
Today at 9:46 AM
Point within rectangle with integer distances to vertices.
Goutham   1
N Dec 8, 2010 by oneplusone
The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.
1 reply
Goutham
Dec 7, 2010
oneplusone
Dec 8, 2010
Point within rectangle with integer distances to vertices.
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Goutham
3130 posts
#1 • 2 Y
Y by Adventure10, Mango247
The lengths of the sides of a rectangle are given to be odd integers. Prove that there does not exist a point within that rectangle that has integer distances to each of its four vertices.
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oneplusone
1459 posts
#2 • 1 Y
Y by Adventure10
Suppose the rectangle has side lengths $a,b$. Let the rectangle be $ABCD$ with $AB=b$. Let $P$ be the point inside. Let $X,Y$ be the feet of the perpendiculars from $P$ onto $CD,DA$. Let $DX=x$. Then $x^2-(b-x)^2=PD^2-PC^2$ is an integer, so $b(2x-b)$ is an integer, so $x$ is rational. Now let $x=\frac{c}{d}$. Similarly let $DY=\frac{e}{f}$, where $\gcd(c,d)=\gcd(e,f)=1$. Then since $\frac{c^2}{d^2}+\frac{e^2}{f^2}$ is an integer, we must have $d=f$. So now $\frac{\sqrt{c^2+e^2}}{d}$ is an integer. Since the sum of two odd squares cannot be a square, one of $c,e$ must be even; WLOG $c$ is even, so $d$ is odd since they are coprime. If $e$ is even, we consider the length $PB=\frac{\sqrt{(bd-c)^2+(ad-e)^2}}{d}$, which is not an integer since both terms inside the square root are odd. If $e$ is odd, we consider the length $PC=\frac{\sqrt{(bd-c)^2+e^2}}{d}$, again not an integer. Thus there is no such point $P$.
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